IndisputableMonolith.RRF.Hypotheses.EightTick

   1import Mathlib.Data.Fin.Basic
   2import Mathlib.Data.ZMod.Basic
   3import Mathlib.Data.Real.Basic
   4
   5/-!
   6# RRF Hypotheses: 8-Tick Discretization
   7
   8The 8-tick hypothesis: time/process is discretized into 8-beat cycles.
   9
  10This is an EXPLICIT HYPOTHESIS about observed traces, not a definitional axiom.
  11LNAL bytecode uses 8-phase cycles; this module provides the testing interface.
  12
  13## The Hypothesis
  14
  15Observed folding/recognition traces exhibit 8-phase periodicity:
  16- Phase 0-3: LOCK (structure formation)
  17- Phase 4-7: BALANCE (equilibration)
  18
  19## Falsification Criteria
  20
  211. Observed traces with non-8 periodicity that work better
  222. Traces where 8-phase segmentation doesn't capture dynamics
  233. LNAL programs that require different cycle lengths
  24-/
  25
  26
  27namespace IndisputableMonolith
  28namespace RRF.Hypotheses
  29
  30/-! ## The 8-Tick Structure (Definitional) -/
  31
  32/-- A phase in the 8-beat cycle. -/
  33abbrev Phase := Fin 8
  34
  35namespace Phase
  36
  37/-- Phase 0: Start of LOCK. -/
  38def lock_start : Phase := 0
  39/-- Phase 3: End of LOCK. -/
  40def lock_end : Phase := 3
  41/-- Phase 4: Start of BALANCE. -/
  42def balance_start : Phase := 4
  43/-- Phase 7: End of BALANCE. -/
  44def balance_end : Phase := 7
  45
  46/-- Is this phase in the LOCK region? -/
  47def isLock (p : Phase) : Bool := p.val ≤ 3
  48
  49/-- Is this phase in the BALANCE region? -/
  50def isBalance (p : Phase) : Bool := p.val ≥ 4
  51
  52/-- LOCK and BALANCE partition the phases. -/
  53theorem lock_balance_partition (p : Phase) : p.isLock ∨ p.isBalance := by
  54  simp [isLock, isBalance]
  55  omega
  56
  57/-- LOCK and BALANCE are disjoint. -/
  58theorem lock_balance_disjoint (p : Phase) : ¬(p.isLock ∧ p.isBalance) := by
  59  simp [isLock, isBalance]
  60  omega
  61
  62/-- Successor phase (cyclic). -/
  63def next (p : Phase) : Phase := p + 1
  64
  65/-- Predecessor phase (cyclic). -/
  66def prev (p : Phase) : Phase := p - 1
  67
  68theorem next_prev (p : Phase) : (p.next).prev = p := by
  69  simp [next, prev]
  70
  71theorem prev_next (p : Phase) : (p.prev).next = p := by
  72  simp [next, prev]
  73
  74end Phase
  75
  76/-! ## The 8-Tick Hypothesis Class -/
  77
  78/-- An observed trace with tick structure. -/
  79structure TickedTrace (Event : Type*) where
  80  /-- The events in sequence. -/
  81  events : List Event
  82  /-- The phase of each event. -/
  83  phase : Fin events.length → Phase
  84
  85/-- The 8-tick hypothesis for traces.
  86
  87This is the explicit claim that traces exhibit 8-phase structure.
  88-/
  89class EightTickHypothesis (Trace : Type*) where
  90  /-- Extract tick information from a trace. -/
  91  ticks : Trace → List Phase
  92  /-- The tick sequence respects the 8-cycle. -/
  93  cyclic : ∀ t : Trace, ∀ i : Nat, ∀ hi : i < (ticks t).length,
  94    let next_i := (i + 1) % (ticks t).length
  95    ∀ hn : next_i < (ticks t).length,
  96    (ticks t).get ⟨i, hi⟩ = (ticks t).get ⟨next_i, hn⟩ ∨
  97    ((ticks t).get ⟨i, hi⟩).next = (ticks t).get ⟨next_i, hn⟩
  98
  99/-- Prediction obligation: LOCK phases should show structural change. -/
 100structure LockPhasePrediction (Event : Type*) where
 101  /-- Metric for structural change. -/
 102  structuralChange : Event → ℝ
 103  /-- LOCK phases have higher structural change. -/
 104  lock_higher : ∀ (trace : TickedTrace Event) (i j : Fin trace.events.length),
 105    (trace.phase i).isLock → (trace.phase j).isBalance →
 106    structuralChange (trace.events[i]) ≥ structuralChange (trace.events[j])
 107
 108/-! ## Falsification Interface -/
 109
 110/-- A falsification witness: a trace where 8-tick doesn't work. -/
 111structure EightTickFalsifier (Event : Type*) where
 112  /-- The observed trace. -/
 113  trace : List Event
 114  /-- The optimal periodicity (not 8). -/
 115  optimalPeriod : ℕ
 116  /-- The optimal period is not 8. -/
 117  not_eight : optimalPeriod ≠ 8
 118  /-- Evidence that this period works better (abstract). -/
 119  works_better : True  -- Placeholder for a concrete metric
 120
 121/-- Alternative: 4-tick (half cycle) or 16-tick (double cycle) might also work. -/
 122def validAlternativePeriods : List ℕ := [4, 8, 16]
 123
 124/-- If a falsifier exists with period outside valid alternatives, it's strong evidence. -/
 125def strongFalsifier (E : Type*) (f : EightTickFalsifier E) : Prop :=
 126  f.optimalPeriod ∉ validAlternativePeriods
 127
 128end RRF.Hypotheses
 129end IndisputableMonolith
 130